Elementary properties of Complex numbers notes - Complex Uploaded By Kiranmayi. Complex Analysis Complex analysis is the theory of functions of complex numbers, like real analysis is the theory of functions of real numbers.
Why need Complex Analysis? Evaluation of many real integrals: e. Solving Differential Equations. In Fourier Analysis. In Number Theory. All major branches of Mathematics which are applicable in science and engineering. Lecture 1 Complex Analysis. It has no real root. Assume that i iota is a solution. So we define it as imaginary number. It is not immediately obvious that this extension is in fact algebraically closed; this is the content of the famous fundamental theorem of algebra , which we will prove later in this course. The two equivalent definitions of — as the algebraic closure, and as a quadratic extension, of the reals respectively — each reveal important features of the complex numbers in applications.
Because is algebraically closed, all polynomials over the complex numbers split completely, which leads to a good spectral theory for both finite-dimensional matrices and infinite-dimensional operators; in particular, one expects to be able to diagonalise most matrices and operators. Applying this theory to constant coefficient ordinary differential equations leads to a unified theory of such solutions, in which real-variable ODE behaviour such as exponential growth or decay, polynomial growth, and sinusoidal oscillation all become aspects of a single object, the complex exponential or more generally, the matrix exponential.
Applying this theory more generally to diagonalise arbitrary translation-invariant operators over some locally compact abelian group, one arrives at Fourier analysis , which is thus most naturally phrased in terms of complex-valued functions rather than real-valued ones.
If one drops the assumption that the underlying group is abelian, one instead discovers the representation theory of unitary representations , which is simpler to study than the real-valued counterpart of orthogonal representations. For closely related reasons, the theory of complex Lie groups is simpler than that of real Lie groups. Meanwhile, the fact that the complex numbers are a quadratic extension of the reals lets one view the complex numbers geometrically as a two-dimensional plane over the reals the Argand plane.
Whereas a point singularity in the real line disconnects that line, a point singularity in the Argand plane leaves the rest of the plane connected although, importantly, the punctured plane is no longer simply connected.
As we shall see, this fact causes singularities in complex analytic functions to be better behaved than singularities of real analytic functions, ultimately leading to the powerful residue calculus for computing complex integrals. Another important geometric feature of the Argand plane is the angle between two tangent vectors to a point in the plane. As it turns out, the operation of multiplication by a complex scalar preserves the magnitude and orientation of such angles; the same fact is true for any non-degenerate complex analytic mapping, as can be seen by performing a Taylor expansion to first order.
This fact ties the study of complex mappings closely to that of the conformal geometry of the plane and more generally, of two-dimensional surfaces and domains. In particular, one can use complex analytic maps to conformally transform one two-dimensional domain to another, leading among other things to the famous Riemann mapping theorem , and to the classification of Riemann surfaces.
enter If one Taylor expands complex analytic maps to second order rather than first order, one discovers a further important property of these maps, namely that they are harmonic. This fact makes the class of complex analytic maps extremely rigid and well behaved analytically; indeed, the entire theory of elliptic PDE now comes into play, giving useful properties such as elliptic regularity and the maximum principle.
In fact, due to the magic of residue calculus and contour shifting, we already obtain these properties for maps that are merely complex differentiable rather than complex analytic, which leads to the striking fact that complex differentiable functions are automatically analytic in contrast to the real-variable case, in which real differentiable functions can be very far from being analytic. The geometric structure of the complex numbers and more generally of complex manifolds and complex varieties , when combined with the algebraic closure of the complex numbers, leads to the beautiful subject of complex algebraic geometry , which motivates the much more general theory developed in modern algebraic geometry.
However, we will not develop the algebraic geometry aspects of complex analysis here. Last, but not least, because of the good behaviour of Taylor series in the complex plane, complex analysis is an excellent setting in which to manipulate various generating functions , particularly Fourier series which can be viewed as boundary values of power or Laurent series , as well as Dirichlet series.
The theory of contour integration provides a very useful dictionary between the asymptotic behaviour of the sequence , and the complex analytic behaviour of the Dirichlet or Fourier series, particularly with regard to its poles and other singularities. This turns out to be a particularly handy dictionary in analytic number theory , for instance relating the distribution of the primes to the Riemann zeta function.
We will frequently touch upon many of these connections to other fields of mathematics in these lecture notes. However, these are mostly side remarks intended to provide context, and it is certainly possible to skip most of these tangents and focus purely on the complex analysis material in these notes if desired. Note: complex analysis is a very visual subject, and one should draw plenty of pictures while learning it. Note: this section will be far more algebraic in nature than the rest of the course; we are concentrating almost all of the algebraic preliminaries in this section in order to get them out of the way and focus subsequently on the analytic aspects of the complex numbers.
Thanks to the laws of high-school algebra, we know that the real numbers are a field : it is endowed with the arithmetic operations of addition, subtraction, multiplication, and division, as well as the additive identity and multiplicative identity , that obey the usual laws of algebra i. The algebraic structure of the reals does have one drawback though — not all non-trivial polynomials have roots! Most famously, the polynomial equation has no solutions over the reals, because is always non-negative, and hence is always strictly positive, whenever is a real number.
Step 1: Convert to exponential form. Some of these applications are described below. Exercise 16 Let be a sequence of complex numbers. Similarly, using Cartesian coordinates, we see that the operation of adding a complex number by a given complex number is simply a spatial translation by a displacement of. Observe that the standard branch of the argument has a discontinuity on the negative real axis , which is the branch cut of this branch. Note: The graphic calculator is unable to solve polynomial equations with complex coefficients. In fact, there is no linear ordering on the complex numbers that is compatible with addition and multiplication — the complex numbers cannot have the structure of an ordered field.
As mentioned in the introduction, one traditional way to define the complex numbers is as the smallest possible extension of the reals that fixes this one specific problem:. Definition 1 The complex numbers A field of complex numbers is a field that contains the real numbers as a subfield, as well as a root of the equation.
Thus, strictly speaking, a field of complex numbers is a pair , but we will almost always abuse notation and use as a metonym for the pair. Furthermore, is generated by and , in the sense that there is no subfield of , other than itself, that contains both and ; thus, in the language of field extensions , we have. We will take the existence of the real numbers as a given in this course; constructions of the real number system can of course be found in many real analysis texts, including my own.
Definition 1 is short, but proposing it as a definition of the complex numbers raises some immediate questions:. The third set of questions can be answered satisfactorily once we possess the fundamental theorem of algebra. For now, we focus on the first two questions. We begin with existence. One can construct the complex numbers quite explicitly and quickly using the Argand plane construction; see Remark 7 below.
However, from the perspective of higher mathematics, it is more natural to view the construction of the complex numbers as a special case of the more general algebraic construction that can extend any field by the root of an irreducible nonlinear polynomial over that field; this produces a field of complex numbers when specialising to the case where and. We will just describe this construction in that special case, leaving the general case as an exercise.
Starting with the real numbers , we can form the space of formal polynomials.
A small technical point: we do not view this indeterminate as belonging to any particular domain such as , so we do not view these polynomials as functions but merely as formal expressions involving a placeholder symbol which we have rendered in Roman type to indicate its formal character. In this particular characteristic zero setting of working over the reals, it turns out to be harmless to identify each polynomial with the corresponding function formed by interpreting the indeterminate as a real variable; but if one were to generalise this construction to positive characteristic fields, and particularly finite fields, then one can run into difficulties if polynomials are not treated formally, due to the fact that two distinct formal polynomials might agree on all inputs in a given finite field e.
However, this subtlety can be ignored for the purposes of this course. This space of polynomials has a pretty good algebraic structure, in particular the usual operations of addition, subtraction, and multiplication on polynomials, together with the zero polynomial and the unit polynomial , give the structure of a unital commutative ring. This commutative ring also contains as a subring identifying each real number with the degree zero polynomial.
The ring is however not a field, because many non-zero elements of do not have multiplicative inverses. In fact, no non-constant polynomial in has an inverse in , because the product of two non-constant polynomials has a degree that is the sum of the degrees of the factors. If a unital commutative ring fails to be field, then it will instead possess a number of non-trivial ideals.
The only ideal we will need to consider here is the principal ideal. This is clearly an ideal of — it is closed under addition and subtraction, and the product of any element of the ideal with an element of the full ring remains in the ideal. We now define to be the quotient space. Because is an ideal, there is an obvious way to define addition, subtraction, and multiplication in , namely by setting.
Also, the real line embeds into by identifying each real number with the coset ; note that this identification is injective, as no real number is a multiple of the polynomial. If we define to be the coset.
Thus contains both and a solution of the equation. Also, since every element of is of the form for some polynomial , we see that every element of is a polynomial combination of with real coefficients; in particular, any subring of that contains and will necessarily have to contain every element of. Thus is generated by and. The only remaining thing to verify is that is a field and not just a commutative ring. In other words, we need to show that every non-zero element of has a multiplicative inverse.
This stems from a particular property of the polynomial , namely that it is irreducible in. That is to say, we cannot factor into non-constant polynomials. Indeed, as has degree two, the only possible way such a factorisation could occur is if both have degree one, which would imply that the polynomial has a root in the reals , which of course it does not. Because the polynomial is irreducible, it is also prime : if divides a product of two polynomials in , then it must also divide at least one of the factors ,. Indeed, if does not divide , then by irreducibility the greatest common divisor of and is.
Applying the Euclidean algorithm for polynomials, we then obtain a representation of as. Since is prime, the quotient space is an integral domain : there are no zero-divisors in other than zero. This brings us closer to the task of showing that is a field, but we are not quite there yet; note for instance that is an integral domain, but not a field.
But one can finish up by using finite dimensionality. As is a ring containing the field , it is certainly a vector space over ; as is generated by and , and , we see that it is in fact a two-dimensional vector space over , spanned by and which are linearly independent, as clearly cannot be real. In particular, it is finite dimensional. For any non-zero , the multiplication map is an -linear map from this finite-dimensional vector space to itself.
LECTURE NOTES, SPRING BJORN POONEN. 7. Complex numbers . Complex numbers are expressions of the form x + yi, where x and y are real. faculty of mathematical studies mathematics for part engineering lectures module 22 complex numbers ii revision relationships between trigonometric and.
As is an integral domain, this map is injective; by finite-dimensionality, it is therefore surjective by the rank-nullity theorem. In particular, there exists such that , and hence is invertible and is a field. This concludes the construction of a complex field. Remark 2 One can think of the action of passing from a ring to a quotient by some ideal as the action of forcing some relations to hold between the various elements of , by requiring all the elements of the ideal or equivalently, all the generators of to vanish.
Thus one can think of as the ring formed by adjoining a new element to the existing ring and then demanding the constraint. With this perspective, the main issues to check in order to obtain a complex field are firstly that these relations do not collapse the ring so much that two previously distinct elements of become equal, and secondly that all the non-zero elements become invertible once the relations are imposed, so that we obtain a field rather than merely a ring or integral domain.
Remark 3 It is instructive to compare the complex field , formed by adjoining the square root of to the reals, with other commutative rings such as the dual numbers which adjoins an additional square root of to the reals or the split complex numbers which adjoins a new root of to the reals.
The latter two objects are perfectly good rings, but are not fields they contain zero divisors, and the first ring even contains a nilpotent. This is ultimately due to the reducible nature of the polynomials and in.